Ingegneria Civile per l'Ambiente ed il Territorio | MATHEMATICS I

cod. 0612500001

MATHEMATICS I

	0612500001
	DIPARTIMENTO DI INGEGNERIA CIVILE
	EQF6
	CIVIL AND ENVIRONMENTAL ENGINEERING
	2018/2019

PERCORSO COMUNE

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2018
	PRIMO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/05	9	90	LESSONS	BASIC COMPULSORY SUBJECTS

PROFESSORS

	MICHELE CIARLETTA T Curriculum
	FEDERICA GREGORIO Curriculum

	Objectives
	THE COURSE WILL FURNISH THE FIRST MATHEMATICAL BASES TO FACE THE NEXT COURSES. IT AIMS AT THE FOLLOWING EDUCATIONAL OBJECTIVES: -) KNOWLEDGE AND UNDERSTANDING THE STUDENT HAS TO LEARN THE FUNDAMENTAL FACTS OF MATHEMATICAL ANALYSIS, IN PARTICULAR SETS OF NUMBERS, REAL-VALUED FUNCTIONS, SEQUENCES OF REAL NUMBERS, LIMITS OF REAL-VALUED FUNCTIONS, CONTINUOUS FUNCTIONS, DERIVATIVES OF REAL-VALUED FUNCTIONS, THE FUNDAMENTAL THEORY OF DIFFERENTIAL CALCULATION, THE GRAPHIC STUDY OF A FUNCTION, THE INTEGRALS OF THE FUNCTIONS OF A VARIABLE AND NUMERICAL SERIES OF REAL NUMBERS. -) APPLYING KNOWLEDGE AND UNDERSTANDING THE STUDENT HAS TO DEVELOP IN A RIGOROUS AND COHERENT WAY A MATHEMATICAL ARGUMENT, APPLY THE THEOREMES AND THE STUDIED RULES TO SOLVING PROBLEMS. HE HAS TO BE ABLE TO MAKE CALCULATIONS WITH LIMITS, DERIVATIVES AND INTEGRALS (BOTH UNDEFINED AND DEFINED). -) LEARNING SKILLS THE STUDENT HAS TO DEVELOP THE LEARNING SKILLS THAT WILL BE NECESSARY FOR INSERTING HIM IN THE FOLLOWING STUDIES WITH A HIGH AUTONOMY OF STUDY, AND CRITICALLY FACE MORE GENERAL PROBLEMS.

THE COURSE WILL FURNISH THE FIRST MATHEMATICAL BASES TO FACE THE NEXT COURSES. IT AIMS AT THE FOLLOWING EDUCATIONAL OBJECTIVES:

-) KNOWLEDGE AND UNDERSTANDING

THE STUDENT HAS TO LEARN THE FUNDAMENTAL FACTS OF MATHEMATICAL ANALYSIS, IN PARTICULAR SETS OF NUMBERS, REAL-VALUED FUNCTIONS, SEQUENCES OF REAL NUMBERS, LIMITS OF REAL-VALUED FUNCTIONS, CONTINUOUS FUNCTIONS, DERIVATIVES OF REAL-VALUED FUNCTIONS, THE FUNDAMENTAL THEORY OF DIFFERENTIAL CALCULATION, THE GRAPHIC STUDY OF A FUNCTION, THE INTEGRALS OF THE FUNCTIONS OF A VARIABLE AND NUMERICAL SERIES OF REAL NUMBERS.

-) APPLYING KNOWLEDGE AND UNDERSTANDING

THE STUDENT HAS TO DEVELOP IN A RIGOROUS AND COHERENT WAY A MATHEMATICAL ARGUMENT, APPLY THE THEOREMES AND THE STUDIED RULES TO SOLVING PROBLEMS. HE HAS TO BE ABLE TO MAKE CALCULATIONS WITH LIMITS, DERIVATIVES AND INTEGRALS (BOTH UNDEFINED AND DEFINED).

-) LEARNING SKILLS

THE STUDENT HAS TO DEVELOP THE LEARNING SKILLS THAT WILL BE NECESSARY FOR INSERTING HIM IN THE FOLLOWING STUDIES WITH A HIGH AUTONOMY OF STUDY, AND CRITICALLY FACE MORE GENERAL PROBLEMS.

	Prerequisites
	FOR THE SUCCESSFUL ACHIEVEMENT OF THE GOALS, STUDENTS ARE REQUIRED TO MASTER THE FOLLOWING PREREQUISITES: -KNOWLEDGE OF ALGEBRA, WITH PARTICULAR REFERENCE TO: ALGEBRAIC EQUATIONS AND INEQUALITIES, LOGARITHMIC, EXPONENTIAL, TRIGONOMETRIC, TRANSCENDENTAL, -KNOWLEDGE RELATED TO TRIGONOMETRY, WITH PARTICULAR REFERENCE TO THE BASIC TRIGONOMETRIC FUNCTIONS. NO PROPEDEUTIC COURSES ARE NEEDED.

	Contents
	NUMERICAL SETS: INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. INTRODUCTION TO REAL NUMBERS. EXTREME VALUES. INTERVALS OF REAL NUMBERS. NEIGHBORHOODS, ACCUMULATION POINTS. CLOSED AND OPEN SETS. INTRODUCTION TO COMPLEX NUMBERS. OPERATIONS ON COMPLEX NUMBERS. POWERS AND DE MOIVRE’SFORMULA. N-THROOTS.(HOURS 5/3/-) REAL FUNCTIONS:DEFINITION. DOMAIN, CODOMAIN AND GRAPH. EXTREMA. MONOTONE, COMPOSITE INVERTIBLE FUNCTIONS.ELEMENTARY FUNCTIONS: N-THPOWER AND ROOT, EXPONENTIAL, LOGARITHMIC, POWER, TRIGONOMETRIC AND INVERSE FUNCTIONS.(HOURS 4/2/-) BASIC NOTIONS OF EQUATIONS AND INEQUALITIES: FIRST ORDER, QUADRATIC, BINOMIAL,IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC EQUATIONS. SYSTEMS. FIRST ORDER,QUADRATIC, RATIONAL, IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC INEQUALITIES. SYSTEMS.(HOURS 2/3/-) NUMERICAL SEQUENCES:DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. EULER’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE.(HOURS 2/2/-) LIMITS OF A FUNCTION: DEFINITION. RIGHTAND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS.(HOURS 5/3/-) CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY.(HOURS 5/-/-) DERIVATIVE OF A FUNCTION: DEFINITION. LEFT AND RIGHT DERIVATIVES. GEOMETRIC MEANING. DIFFERENTIABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY, COMPOSITE, INVERSE FUNCTIONS. HIGHER ORDER DERIVATIVES. DIFFERENTIAL OF A FUNCTION AND ITS GEOMETRIC MEANING.(HOURS 5/3/-) FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITALTHEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS.(HOURS 4/3/-) GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. FINDING LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAQWING GRAPH.(HOURS 6/8/-). INTEGRATION OF FUNCTIONS OF ONE VARIABLE:DEFINITION OF PRIMITIVE FUNCTION AND INDEFINITE INTEGRAL. IMMEDIATE INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND ITS GEOMETRIC MEANING. MEAN VALUE THEOREM. INTEGRAL FUNCTION AND THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS.(HOURS 6/6/-) NUMERICAL SERIES: INTRODUCTION TO NUMERICAL SERIES.

Contents

NUMERICAL SETS: INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. INTRODUCTION TO REAL NUMBERS. EXTREME VALUES. INTERVALS OF REAL NUMBERS. NEIGHBORHOODS, ACCUMULATION POINTS. CLOSED AND OPEN SETS. INTRODUCTION TO COMPLEX NUMBERS. OPERATIONS ON COMPLEX NUMBERS. POWERS AND DE MOIVRE’SFORMULA. N-THROOTS.(HOURS 5/3/-)
REAL FUNCTIONS:DEFINITION. DOMAIN, CODOMAIN AND GRAPH. EXTREMA. MONOTONE, COMPOSITE INVERTIBLE FUNCTIONS.ELEMENTARY FUNCTIONS: N-THPOWER AND ROOT, EXPONENTIAL, LOGARITHMIC, POWER, TRIGONOMETRIC AND INVERSE FUNCTIONS.(HOURS 4/2/-)
BASIC NOTIONS OF EQUATIONS AND INEQUALITIES: FIRST ORDER, QUADRATIC, BINOMIAL,IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC EQUATIONS. SYSTEMS. FIRST ORDER,QUADRATIC, RATIONAL, IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC INEQUALITIES. SYSTEMS.(HOURS 2/3/-)
NUMERICAL SEQUENCES:DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. EULER’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE.(HOURS 2/2/-)
LIMITS OF A FUNCTION: DEFINITION. RIGHTAND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS.(HOURS 5/3/-)
CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY.(HOURS 5/-/-)
DERIVATIVE OF A FUNCTION: DEFINITION. LEFT AND RIGHT DERIVATIVES. GEOMETRIC MEANING. DIFFERENTIABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY, COMPOSITE, INVERSE FUNCTIONS. HIGHER ORDER DERIVATIVES. DIFFERENTIAL OF A FUNCTION AND ITS GEOMETRIC MEANING.(HOURS 5/3/-)
FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITALTHEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS.(HOURS 4/3/-)
GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. FINDING LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAQWING GRAPH.(HOURS 6/8/-).
INTEGRATION OF FUNCTIONS OF ONE VARIABLE:DEFINITION OF PRIMITIVE FUNCTION AND INDEFINITE INTEGRAL. IMMEDIATE INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND ITS GEOMETRIC MEANING. MEAN VALUE THEOREM. INTEGRAL FUNCTION AND THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS.(HOURS 6/6/-)
NUMERICAL SERIES: INTRODUCTION TO NUMERICAL SERIES.

	Teaching Methods
	IT IS A 9-AEC COURSE. THE COURSE COVERS THEORETICAL LECTURES, DEVOTED TO THE FACE-TO-FACE DELIVERY OF ALL THE COURSE CONTENTS , AND CLASSROOM PRACTICE DEVOTED TO PROVIDE THE STUDENTS WITH THE MAIN TOOLS NEEDED TO PROBLEM-SOLVING ACTIVITIES. THE 70 PERCENTOF TOTAL PRESENCES IS REQUIRED.

	Verification of learning
	THE FINAL EXAM CARRIED OUT AT THE END OF THE COURSE AND IT IS DESIGNED TO EVALUATE AS A WHOLE: •THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE COURSE; •THE MASTERY OF THE MATHEMATICAL LANGUAGE IN THE WRITTEN AND ORAL PROOFS; •THE SKILL OF PROVING THEOREMS; •THE SKILL OF SOLVING EXERCISES; •THE SKILL TO IDENTIFY AND APPLY THE BEST AND EFFICIENT METHOD IN EXERCISES SOLVING; •THE ABILITY TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE COURSE. THE EXAM CONSISTS OF A WRITTEN PROOF AND AN ORAL INTERVIEW. WRITTEN PROOF: THE WRITTEN PROOF LASTS 2,5 HOURS AND CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE. IN THE CASE OF A SUFFICIENT PROOF, IT WILL BE EVALUATED BY THREE SCALES. ORAL INTERVIEW: THE INTERVIEW LASTS ABOUT 30 MINUTES AND IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE COURSE, AND COVERS DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING. FINAL EVALUATION: THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE MARK OF THE WRITTEN PROOF, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW. MASTERING ABILITY OF THE COURSE CONTENTS, TAKING INTO ACCOUNT THE QUALITY OF THE WRITTEN AND ORAL ELABORATION AND THE SELF-EVALUATION CAPABILITY SHOWN. LAUDE FOLLOWS FROM BRILLIANT WRITTEN PROOF AND ORAL INTERVIEW.

Verification of learning

THE FINAL EXAM CARRIED OUT AT THE END OF THE COURSE AND IT IS DESIGNED TO EVALUATE AS A WHOLE:
•THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE COURSE;
•THE MASTERY OF THE MATHEMATICAL LANGUAGE IN THE WRITTEN AND ORAL PROOFS;
•THE SKILL OF PROVING THEOREMS;
•THE SKILL OF SOLVING EXERCISES;
•THE SKILL TO IDENTIFY AND APPLY THE BEST AND EFFICIENT METHOD IN EXERCISES SOLVING;
•THE ABILITY TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE COURSE.
THE EXAM CONSISTS OF A WRITTEN PROOF AND AN ORAL INTERVIEW.
WRITTEN PROOF: THE WRITTEN PROOF LASTS 2,5 HOURS AND CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE. IN THE CASE OF A SUFFICIENT PROOF, IT WILL BE EVALUATED BY THREE SCALES.
ORAL INTERVIEW: THE INTERVIEW LASTS ABOUT 30 MINUTES AND IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE COURSE, AND COVERS DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING.
FINAL EVALUATION: THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE MARK OF THE WRITTEN PROOF, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.
MASTERING ABILITY OF THE COURSE CONTENTS, TAKING INTO ACCOUNT THE QUALITY OF THE WRITTEN AND ORAL ELABORATION AND THE SELF-EVALUATION CAPABILITY SHOWN.
LAUDE FOLLOWS FROM BRILLIANT WRITTEN PROOF AND ORAL INTERVIEW.

	Texts
	P. MARCELLINI - C. SBORDONE, “ELEMENTI DI CALCOLO “, LIGUORI EDITORE P. MARCELLINI - C. SBORDONE, “ANALISI MATEMATICA UNO “, LIGUORI EDITORE P. MARCELLINI - C. SBORDONE, “ELEMENTI DI ANALISI MATEMATICA UNO “, LIGUORI EDITORE ESERCIZI P. MARCELLINI - C. SBORDONE, “ESERCITAZIONI DI MATEMATICA I “, LIGUORI EDITORE